24 August 2000

Ž .Physics Letters B 488 2000 27–30www.elsevier.nlrlocaternpe

Quantum gauge symmetry from finite field dependentBRST transformations

Rabin Banerjee, Bhabani Prasad MandalS.N. Bose National Center for Basic Sciences, Block-JD; Sector-III; Salt Lake, Calcutta-700 091, India

Received 30 June 2000; accepted 7 July 2000Editor: M. Cvetic

Dedicated to the memory of Prof. Chanchal Kumar Majumdar

Abstract

Using the technique of finite field dependent BRST transformations we show that the classical massive Yang–Millstheory and the pure Yang–Mills theory whose gauge symmetry is broken by a gauge fixing term are identical from the viewpoint of quantum gauge symmetry. The explicit infinitesimal transformations which leave the massive Yang–Mills theoryBRST invariant are given. q 2000 Elsevier Science B.V. All rights reserved.

w xIn a recent paper 1 it was shown that a classicalmassive gauge theory does not have an essentialdifference, at the quantum level, from a gauge in-variant theory whose gauge symmetry is broken by agauge fixing term. Specifically, the classical la-grangians,

m2a aLLsLL y A A , 1Ž .YM m m2

and21 aLLsLL y E A , 2Ž .Ž .YM m m2

where LL is the Yang–Mills lagrangian,YM

21 a a abc b cLL sy E A yE A qg f A A , 3Ž .Ž .YM m n n m m n4

Ž .E-mail addresses: rabin@boson.bose.res.in R. Banerjee ,Ž .bpm@boson.bose.res.in B.P. Mandal .

could be given an identical physical meaning, bothrepresenting an effective gauge fixed lagrangian as-sociated with the quantum theory defined by

a a a aDDA DDB DDc DDc exp ySH m YM½a4 a m a a mq d x yiB E A qc yE D c ,Ž .Ž .H ž /m m 5

4Ž .

that is invariant under the BRST transformations,

gaa a abc b cd A syi D c dl , dc syi f c c dl ,Ž .m m 2a a adc sB dl , dB s0 , 5Ž .

where dl is an infinitesimal Grassmann parameter.This is to be contrasted with the conventional inter-

Ž .pretation of regarding 1 as a massive vector theoryŽ .and 2 as an effective Yang–Mills theory with a

covariant gauge fixing term.

0370-2693r00r$ - see front matter q 2000 Elsevier Science B.V. All rights reserved.Ž .PII: S0370-2693 00 00842-X

( )R. Banerjee, B.P. MandalrPhysics Letters B 488 2000 27–3028

In this paper we shall show the equivalence of theŽ . Ž .quantum theories defined by 1 and 2 by following

Ž .the method of finite field dependent BRST FFBRSTw xtransformations developed by one of us 2 . In partic-

ular this method, which will be briefly reviewedbelow, connects quantum gauge theories in differentgauges. Here we start from the conventional gauge

Ž .fixed Yang–Mills lagrangian defined by 2 . Theexplicit FFBRST transformations are then statedwhich maps this theory to one whose lagrangian is

Ž .defined by 1 , thereby showing the connection be-tween them. We also get the form of the transforma-tions that preserve the BRST invariance of the quan-

Ž .tum theory defined by 1 . Finally we suggest apossible connection between our approach and that

w xadopted in Ref. 1 , which was based on a modifiedw xquantization scheme 3,4 , where the variation of the

gauge field in the path integral is taken over theentire gauge orbit.

Let us now briefly review the FFBRST approachw x2,5–7 . FFBRST transformations are obtained by an

Ž .integration of infinitesimal field dependent BRSTw xtransformations 2 . In this method all the fields are

function of some parameter, k :0FkF1. For aŽ . Ž . Ž .generic field f x,k , f x,k s 0 s f x and

Ž . XŽ .f x,ks1 sf x . Then the infinitesimal field de-pendent BRST transformations are defined as

dX

f x ,k sd f x ,k Q f x ,k , 6Ž . Ž . Ž . Ž .BRSTdk

where QXdk is an infinitesimal field dependent pa-

rameter. It has been shown by integrating theseXŽ .equations from ks0 to ks1 that f x are related

Ž .to f x by FFBRST,X

f x sf x qd f x Q f x , 7Ž . Ž . Ž . Ž . Ž .BRST

w Ž .x Xw Ž .xwhere Q f x is obtained from Q f x throughthe relation

exp f f x y1Ž .X

Q f x sQ f x , 8Ž . Ž . Ž .f f xŽ .

XŽ .dQ x Ž .and f is given by fsÝ d f xi BRST iŽ .df xi

The choice of the parameter QX is crucial in

connecting different effective gauge theories bymeans of the FFBRST. In particular the FFBRST of

X a aŽ . w Ž .x Ž . w Ž .xwEq. 7 with Q f x,k s iHc y F A kX aw Ž .x xyF A k relates the Yang–Mills theory with an

w xarbitrary gauge fixing F A to the Yang–Mills the-Xw x w xory with another arbitrary gauge fixing F A 5 .

The meaning of these field transformations is asfollows. We consider the vacuum expectation value

w xof a gauge invariant functional G f in some arbi-w xtrary gauge F A ,

²² :: Fw x w x w xG f ' DDfG f exp iS f , 9Ž .Ž .H eff

where

1F 4 2 4 a ab bw xS sS y d xF A y d xc W c , 10Ž .H Heff 0 2

with

dF aab cb w xW s D A . 11Ž .mcd Am

Here S is the pure Yang–Mills action obtained from0Ž . abw x ab3 and the covariant derivative, D A 'd E qm m

g f abcAc . For simplicity we have set the gauge pa-m1 4 2w xrameter ls1 in the gauge fixing term Hd xF A .2l

Now we perform the FFBRST transformationsX Ž .f™f given by 7 . We have then

²² :: ²² X :: X X Xw x w x w x w xG f s G f s DDf J f G fH= F w X xexp iS f , 12Ž .Ž .eff

on account of BRST invariance of S F and gaugeeffw x w X xinvariance of G f . Here J f is the Jacobian

associated with FFBRST and defined asX w X xDDfsDDf J f , 13Ž .

w x w xAs shown in Ref. 2 for the special case G f s1w X x Ž .the Jacobian J f in Eq. 12 can always be re-

Ž w X x.placed by exp iS f with1

F w X x w X x F X

w X xS f qS f sS f , 14Ž .eff 1 eff

where

X X X1F 4 2 4 a ab bw xS sS y d xF A y d xc W c , 15Ž .H Heff 0 2

with

dF a X

Xab cb w xW s D A . 16Ž .mcd Am

( )R. Banerjee, B.P. MandalrPhysics Letters B 488 2000 27–30 29

The extra piece in the action which arises fromthe Jacobian of such FFBRST is given by

X1 14 2 2w x w x w xS f s d x y F A q F AH1 2 2

Xw xqc WyW c . 17Ž .Ž .Thus the FFBRST in Eq. 7 takes the theory with

gauge F to the corresponding theory with gauge FX.We are now ready to apply this machinery to the

present problem. We start with the generating func-tional for the Yang–Mills theory in the Lorentzgauge,

LZs DDA DDc DDcexp iS , 18Ž .Ž .H m eff

where

21L 4 m 4S sS y d x E A y d xcWc , 19Ž .Ž .H Heff 0 m2

with WsE mD is the Faddeev–Popov determinant.m

w Ž .xWe now apply FFBRST Eq. 7 withmv

X 4 a m a aQ s i d yc y E A ym A y , 20Ž . Ž . Ž .H m m< <v

where v m is an arbitrary 4-vector, to the expressionfor the generating functional to obtain,

X X X LZs DDA DDc DDc expi S qS . 21Ž .Ž .H m eff 1

The additional piece in the action comes from thenon-trivial Jacobian of the FFBRST and can be

Ž .written using Eq. 17

1 2 214 2 m mS s d x y m v A q E AŽ . Ž .H1 m m22< <2 v

Xyc W yW c , 22Ž . Ž .

X v m

with W sm D . Hence we obtain the generating< < mv

functional for a new effective action given bymv

14 mnS sS y d x A M A qcm D c , 23Ž .Heff 0 m n m2 < <v

where M mn is a generalized mass matrix,

v mv n

mn 2M sm . 24Ž .2< <v

Ž .This effective action 23 corresponds to the Yang–Mills lagrangian with a generalized mass term. Itshows the connection between the Lorentz gauge and

1 mna generalized ‘mass’ gauge A M A in the con-m n2

text of Yang–Mills theory. To exactly reproduce thefamiliar mass term, we restrict the arbitrary vectorv m to be of infinitesimal form satisfying the sym-metric multiplication rule,

v mv g mn n

s . 25Ž .2 4< <v

1 2 nIn that case the gauge fixing term is m A Am8

which coincides with the standard mass term, after aproper normalization of m.

The infinitesimal BRST transformations whichŽ .leave the Yang–Mills theory with a mass term 23

invariant are given by

ga ab b a abc b cd A sD c dl , dc sy f c c dl ,m m 2

v m

a adc sm A dl . 26Ž .m< <v

We have shown how, by means of finite fielddependent BRST transformations, it was possible tointerpolate between the Yang–Mills theory in thecovariant gauge to the Yang–Mills theory in a masslike gauge. Since FFBRST also connects the Yang–

w xMills theory in the axial and covariant gauges 5,6 itis clear that the Yang–Mills theory with a mass likegauge fixing term can also be obtained from otherstarting points. In this paper we took the covariantgauge as the starting point for reasons of conve-

w xnience and also comparing our analysis with 1 . Itmay be pointed out that the latter approach is basedon the variation of the gauge variable along theentire gauge orbit, without taking any specific limitof the gauge fixing parameter. Consequently thereseems to be a connection between this approach andthe FFBRST method, which is not altogether surpris-ing. Carrying out the integration over the completegauge orbit would be simulated by finite BRSTtransformations instead of the conventional infinites-imal one. We feel it might be useful to pursue thisconnection in a later work.

( )R. Banerjee, B.P. MandalrPhysics Letters B 488 2000 27–3030

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